The solution of a mildly singular integral equation of the first kind on a ball

Abstract

We consider the equations

$$\int_{\left| y \right| \leqslant 1} {\frac{{F(y)}}{{\left| {x - y} \right|^\lambda }} dy = G(x)(\left| x \right| \leqslant 1)} $$

where x and y ∈ E^p, p ≥ 3 and λ < p. For λ ∈ (p-2,p) we show that this equation has at most one integrable solution which if G is twice differentiable actually exists and is, in fact, given by an explicit formula in terms of integral operators acting on G and its derivatives. When λ ≤ p-2 and λ ≠ −2j (j=0,1,⋯) the equation also has at most one integrable solution for which, assuming it to exist and G to be sufficiently smooth, there also is an explicit formula; in this situation, though, the explicit formula does not usually provide an integrable solution, because, in general, such solutions do not exists when λ ≤ p-2 and λ ≠ −2j (j=0,1,⋯), no matter how smooth G is. In the remaining case λ = −2j (j=0,1,⋯), neither uniqueness nor existence holds for solutions of the equation.